We must find the common difference of the A.P:-
Common difference (d) = first term − second term
\(= 18 \frac 15 - 19\)
\(= \frac{91}5 - \frac{95}5\)
\(= \frac{-4}5\)
Hence,
the common difference is \(\frac{-4}5\) or − 0.8
Now to estimate which term of this A.P will be negative let us assume
the nth term of A.P is 0.
Formula for the nth term of an A.P is = a + (n − 1)d
then,
0 = 19 + (n − 1) × −0.8
0 = 19 − 0.8n + 0.8
0.8n = 19.8
n = \(\frac{19.8}{0.8}\)
n = 24.75
If the 24.75th term is 0. Then surly 25th term of the A.P will be negative.
Let us make sure, just in this case:
25th term of an A.P = a + (n − 1)d
= 19 + (25 − 1) × −0.8
= 19 + (24 × (−0.8))
= 19 + (−19.2)
= 19 − 19.2
= −0.2
Thus, the 25th term of the progression is first negative term.